Weighted a priori estimates for elliptic equations

Maria Eugenia Cejas; Ricardo Duran

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arXiv · 1711.00879

Weighted a priori estimates for elliptic equations

Maria Eugenia Cejas·Ricardo Duran

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Abstract

We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a generalization to bounded domains of the one used in $\mathbb{R}^n$ to prove the continuity of singular integral operators in weighted norms. In the case of singular integral operators it is known that the $A_p$ condition is also necessary for the continuity. We do not know whether this is also true for the a priori estimates in bounded domains but we are able to prove a weaker result when the operator is the Laplacian or a power of it. We prove that a necessary condition is that the weight belongs to the local $A_p$ class.

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Maria Eugenia Cejas, Ricardo Duran. 2017-11-02. Weighted a priori estimates for elliptic equations. https://arxiv.org/abs/1711.00879

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